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We are concerned with the existence and boundary behaviour of positive radialsolutions for the system \begin{equation*} \left\{ \begin{aligned} \Deltau&=|x|^{a}v^{p} &&\quad\mbox{ in } \Omega, \\ \Delta v&=|x|^{b}v^{q}f(|\nablau|) &&\quad\mbox{ in } \Omega, \end{aligned} \right. \end{equation*} where$\Omega \subset \bR^N$ is either a ball centered at the origin or the wholespace $\bR^N$, $a$, $b$, $p$, $q> 0$, and $f \in C^1[0, \infty)$ is anincreasing function such that $f(t)> 0$ for all $t> 0$. Firstly, we study theexistence of positive radial solutions in case when the system is posed in aball corresponding to their behaviour at the boundary. Next, we take $f(t) =t^s$, $s> 1$, $\Omega = \bR^N$ and by the use of dynamical system techniques weare able to describe the behaviour at infinity for such positive radialsolutions.

Existence and boundary behaviour of radial solutions for weighted elliptic systems with gradient terms